1. PROOF BY CONTRADICTION

2. proof by contradiction
Let r be a proposition.
A proof of r by contradiction consists of
proving that not(r) implies a contradiction,
thus concluding that not(r) is false,
which implies that r is true.

3. proof by contradiction
In particular if r is
if p then q
then not(r) is logically equivalent
to p AND (not(q)).
This can be verified by constructing
a truth table.

4. proof by contradiction
So a true proposition
if p then q
may be proved by contradiction as follows:
Assume that p is true and q is false,
and show that this assumption implies
a contradiction.

5. proof by contradiction
One way of proving
that the assumption that
p is true and q is false implies
a contradiction is proving that this
assumption implies that p is false.
Since the same assumption also
implies that p is true, we conclude that
the assumption implies that p is true and p
is false, which is a contradiction.

6. proof by contradiction
EXAMPLE:
Prove that the sum of an even integer
and a non-even integer is non-even.
(Note: a non-even integer is an integer
that is not even.)

7. proof by contradiction
We have to prove that for every even integer
a and every non-even integer b, a+b
is non-even.
This is the same as proving that
For all integers a,b, if [a is even and
b is non-even] then [a+b is non-even].

8. proof by contradiction
We prove that by contradiction.
Assume that
[a is even and b is non-even],
and that [a+b is even]. So for some
integers m,n, a=2m and a+b=2n.
Since b=(a+b)-a, b=2n-2m=2(n-m).
We conclude that b is even. This leads
to a contradiction, since we assumed that
b is non-even.